Mathematical Methods for Robust and Nonlinear Control: Epsrc Summer School

Matthew C. Turner, Declan G. Bates9781848000247, 1-84800-024-3

The underlying theory on which much modern robust and nonlinear control is based can be difficult for both students and industrial practitioners to grasp. In particular, certain mathematical aspects of advanced control theory can be problematic for students coming from a standard engineering background. The Engineering and Physical Sciences Research Council (EPSRC) sponsored Summer School which was held in Leicester UK in September 2006 attempted to “fill in the gap” in students’ appreciation of the theory relevant to several important areas of control. Mathematical Methods for Robust and Nonlinear Control is a collection of lecture notes which were presented at that workshop by a group of experts in advanced control engineering of international origins and consists, broadly, of two parts.The first ten chapters are devoted to the mathematical theory behind several areas of robust and nonlinear control and are aimed at introducing fundamental concepts to the reader. The last six chapters contain detailed case studies which aim to demonstrate the use and effectiveness of these modern techniques in real engineering applications. It is hoped that this book will provide a useful introduction for students of many of the more common robust and nonlinear control techniques as well as serving as a valuable reference for the more adept practitioner.

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Table of contents :
front-matter.pdf……Page 1
Contents……Page 6
Introduction……Page 16
Design Specifications and Fundamental Trade-Offs……Page 18
Frequency Domain Design Specifications and Fundamental Trade-Offs……Page 19
Formulating the Problem……Page 21
Weighting Function Selection……Page 23
Solution of the $H_infty$ Control Problem……Page 24
Design Example: Control Law Design for the Bell 205 Helicopter……Page 26
Fundamental Trade- ffs in Terms of $L$……Page 37
The $H_infty$ Loop-Shaping Design Procedure……Page 39
Advantages of $H_infty$ Loop- Shaping……Page 42
Design Example: Control Law Design for the Harrier V/STOL Aircraft……Page 46
References……Page 56
Structural Methods for Linear Systems: An Introduction……Page 59
Introduction……Page 60
State Space Descriptions……Page 62
Transfer Function Descriptions……Page 63
Background on Polynomial Matrices and Matrix Pencils……Page 64
Matrix Divisors and Minimal Bases……Page 65
Solution of State Space Equations……Page 67
Internal-External and Total Stability……Page 69
Controllability and Observability……Page 70
System Minimality……Page 72
Eigenvalues, Eigenvectors and Free Rectilinear Motions……Page 73
Forced Rectilinear Motions and Frequency Transmission……Page 74
Frequency Transmission Blocking and State Space Zeros……Page 75
Properties of Zero Directions……Page 77
Right Singular Systems……Page 78
Frequency Transmission Blocking for Infinite Frequencies……Page 79
Zero Structure and System Transformations……Page 81
The Zero Pencil of Strictly Proper System……Page 82
Decoupling Zeros……Page 83
Dynamic Characterisation of Transfer Function Poles and Zeros……Page 84
Smith–McMillan Form Characterisation of Poles and Zeros……Page 85
Infinite Poles and Zeros……Page 86
Smith–McMillan Form at Infinity: Infinite Poles and Zeros……Page 87
Impulsive Dynamics and Properties of Infinite Poles and Zeros [57]……Page 88
Characteristic Gain, Frequency Functions……Page 89
Poles and Zeros of the System Algebraic Functions……Page 90
Root Locus and the Output Zeroing Problem……Page 91
Structural Properties of the Feedback Configuration……Page 92
Closed-Loop Performance and the Return Ratio Difference and Sensitivity Matrices……Page 95
Determinantal Assignment Problems……Page 96
The General Determinantal Assignment Problem……Page 100
Grassmann- Plucker Invariants……Page 101
Conclusions……Page 104
References……Page 106
Introduction……Page 111
State- pace Truncation……Page 113
The Truncation Error……Page 115
Singular Perturbation Approximation……Page 116
Model Reduction Motivation……Page 118
Balanced Realization……Page 120
Stability……Page 122
Error Bound for “one-step” Truncation……Page 124
The Error Bound for Balanced Truncation……Page 125
Balanced Singular Perturbation Approximation……Page 128
Notes and References……Page 129
Problems……Page 131
References……Page 134
Introduction to LMI Problems……Page 135
Fundamental LMI Properties……Page 136
Systems of LMIs……Page 137
Types of LMI Problems……Page 138
Change of Variables……Page 140
Congruence Transformation……Page 141
Schur Complement……Page 142
The S-Procedure……Page 143
The Projection Lemma and Finsler’s Lemma……Page 144
$L_2$ Gain……Page 146
Lyapunov Stability for Discrete- ime Systems……Page 147
$l_2$ Gain……Page 148
Sector Boundedness……Page 149
A Slightly More Detailed Example……Page 150
References……Page 153
Input Constraints in Control Systems……Page 155
Constrained System Description……Page 156
Constrained Control and Anti- indup……Page 158
Problems Due to Saturation……Page 160
Clues from Classical Control……Page 161
Definitions of Stability……Page 164
Saturation Modelling……Page 166
The Multivariable Circle Criterion……Page 169
Anti-windup Problem Definition……Page 171
Architecture……Page 172
Full Order Compensators……Page 175
Simple 2nd-Order Example……Page 177
The Constrained System……Page 180
Conclusion……Page 181
Further Reading……Page 182
References……Page 183
Introduction……Page 186
LMI Formulation of Performance Specifications……Page 187
Normalized Left Coprime Factorization for LTI Systems……Page 188
$H_infty$ Synthesis……Page 190
LMI Formulation of the $H_infty$ Loop- haping Controller Synthesis……Page 192
Controller Reconstruction……Page 194
Static $H_infty$ Flight Control System Design for the Bell 205 Helicopter……Page 195
$H_infty$ Loop- haping for Polytopic Systems……Page 197
Left Coprime Factors for Polytopic Systems……Page 198
LMI Conditions……Page 200
Illustrative Example……Page 201
References……Page 203
Introduction……Page 206
Background Concepts in Analysis……Page 208
Ordinary Differential Equations……Page 210
Autonomous Differential Inclusions……Page 212
Barbalat’s Lemma, LaSalle’s Invariance Principle, and Lyapunov Stability……Page 213
Generalizations of Barbalat’s Lemma……Page 217
Nonautonomous Ordinary Differential Equations……Page 222
Autonomous Differential Inclusions……Page 224
References……Page 229
Introduction……Page 231
A Discontinuous Observer……Page 232
Observers with Linear and Discontinuous Injection……Page 235
TheWalcott and Zak Observer……Page 237
A Convex Parameterization……Page 238
Tyre/Road Friction and Vehicle Modelling……Page 242
Observer Design……Page 244
Summary……Page 248
Notes and References……Page 249
References……Page 250
Introduction……Page 253
Model-Following Control System……Page 254
Sliding-Mode……Page 255
Feedback Control……Page 257
Second-Order Example……Page 259
Application Problem……Page 260
Problem Formulation……Page 261
Pade´ Approximations and Time Delay Systems……Page 262
System Centre Method and Sliding-Mode Control……Page 264
Numerical Example and Simulations……Page 265
Feedback by y and Describing Function……Page 267
References……Page 272
Introduction……Page 275
Negative Feedback: Regulation……Page 276
Negative Feedback: Sensitivity and Robustness……Page 281
Positive Feedback: Amplification……Page 285
Positive Feedback: Switching and Memory……Page 286
Positive Feedback: Oscillations……Page 289
References……Page 292
Introduction……Page 297
Dynamic Model of the Distillation Column……Page 298
Uncertainty Modelling……Page 301
Closed-Loop System Performance Specifications……Page 304
Open- oop and Closed- oop System Interconnections……Page 308
Loop Shaping Design……Page 309
μ- Synthesis……Page 314
Nonlinear System Simulation……Page 322
Conclusions……Page 324
References……Page 326
Hard Disk Drive Servo System……Page 336
Derivation of Uncertainty Model……Page 342
Closed- oop System Design Specifications……Page 347
System Interconnections……Page 349
Controller Design in Continuous Time……Page 350
μ- Design……Page 352
$H_infty$ Loop-shaping Design……Page 358
Comparison of Designed Controllers……Page 359
Controller Order Reduction……Page 366
Design of Discrete-time Controller……Page 368
Nonlinear System Simulation……Page 372
Conclusions……Page 375
References……Page 376
Railway Vehicles: Conventional Configuration……Page 379
Suspension Design Requirements……Page 380
Modelling of Suspensions (for Applying Control)……Page 381
Control Concepts……Page 384
Historical Facts on Tilt Control……Page 387
Tilting Vehicle Modelling……Page 388
Tilt Control Requirements and Assessment Approach……Page 391
Conventional Tilt Control……Page 393
Nulling-Type Tilt Via Robust Control Techniques……Page 400
Multi-objective $H$_$infty$/$H_2$ Nulling- Type Control Via LMIs……Page 405
Case Study Remarks……Page 412
Basic Notation……Page 413
References……Page 417
Introduction……Page 418
The Micro-actuator Control Loop andWindup Problems……Page 420
Anti-windup Compensation for Discrete Linear Control Systems……Page 424
Anti- indup Compensation for the Micro- ctuator……Page 429
The Micro-actuator Control Loop as Part of a Hard-Disk-Drive Servo-System……Page 430
References……Page 433
Lotka-Volterra Equations……Page 436
Analysis of Equilibria……Page 438
Optimal Control……Page 440
Linear, Time-Varying Quadratic Optimal Control……Page 441
Immune System Dynamics……Page 443
Optimal Enhancement of the Immune Response……Page 445
Some Practical Considerations……Page 447
References……Page 448
back-matter.pdf……Page 450